Zeno's Arrow and Modern Physics[1]

Review – 1997

In this article, I would like to distinguish between two ways of relating to dynamic processes, and discuss some of the implications of this distinction that concern completely different fields. I will begin by discussing the paradox of the arrow in flight in the words of Zeno of Ilia, which clearly illustrates the basic distinction, following which I will attempt to derive certain elements of Heisenberg's uncertainty principle in quantum physics. We will continue to discuss the problem of the interpretation of time in Einstein's theory of relativity, and we will discuss some of the implications of this distinction in other fields of thought.

A. The paradox of the arrow in flight

The starting point of our study is naturally found in the Iliad in ancient Greece, where Zeno attempts to challenge the concept of movement through several paradoxes.[2] One of those paradoxes, called the arrow-in-flight paradox, can be formulated as follows:

(1) "The arrow flies, in an indivisible moment of its flight it flies and does not fly at the same time. And both claims are correct: It flies, for if it were not flying every moment, its movement would not be performed. And it does not fly, for in an indivisible moment of time it cannot perform any movement. Say from now on: The arrow flies and does not fly at the same time.".[3] Shmuel Hugo Bergman, who cites this formulation as one of the appeals to the law of contradiction, goes on to say that it is the law for any continuous process such as the freezing of water and more.

The paradox can be formulated slightly differently: (2) At each indivisible instant the arrow is located at a definite place in space which may be different from its previous locations. If so, at what moment in time does the arrow change its location? That is, when does it move?

These two formulations seem to be equivalent on the surface. Both assume the theorem that if a body is stationary at a certain moment, then it cannot also be moving at that moment, which is based on the principle of contradiction (although the second formulation does so implicitly). It will be seen later that these are not completely equivalent formulations.

There are attempts to solve the problem using concepts from infinitesimal calculus and the problematic nature of the concept of continuity. In simple terms, this can be expressed as follows: the continuous line does not consist of points but of infinitesimals as small as we like. The time axis is a continuous line. Therefore, it makes no sense to discuss the state of a body at a point in time that is not divisible and does not expand. Bergman's formulation there is very similar: "It is impossible to divide the continuity into indivisible moments." This claim, like the discussion that will follow in the continuation of this article, is formulated in the context of an intuitive approach to the concept of continuity and infinitesimal calculus.[4] In modern mathematical theory there are also descriptions of the sequence as a collection of points that fulfills some properties (the approach of set theory), and in such an approach the solution to Zeno's paradox seems completely implausible. This distinction between the continuous and the discrete has proven to be very fruitful in the mathematical description of processes using the infinitesimal calculus. Still, it seems that the use of this distinction to solve the paradox does not correspond to simple intuition. This feels that the line can be created from discrete points, and also that it is certainly possible to speak of a discrete point in time and the state of a body at such a point. The assertion underlying the infinitesimal calculus only helps us to avoid technical difficulties in the mathematical description of the sequence. That is, the above assertion, which denies the possibility of dividing the sequence into points, does not seem to be an ontological assertion but rather a mathematical assumption that allows us to formally avoid the paradoxes of the sequence.[5] It is therefore found that the philosophical problematic in describing the arrow's flight remains.[6]

Now I will try to present a different solution to the arrow-in-flight paradox. As I will explain later, it will appear that this solution holds new and surprising insights in various fields.

B. The Arrow and the Uncertainty Principle

The assumption implicit in the two above formulations of the arrow-in-flight paradox is the application of the law of contradiction to the concept of motion. Although the statement that a body cannot move and not move at the same time is a logical tautology, we would like to argue that this is not exactly the assumption underlying the paradox. The different formulations of the paradox assume an identity between the statement that "body A has velocity" and the statement that "body A changes its position." My argument is that the term "moves" is used in the above formulations in these two meanings, and that they are different from each other. In a completely parallel way, it is possible to divide the term "standing" into two meanings: the first is "located" in this place, and the second is "at rest" in this place.

Quantum physics faced a dilemma, similar in some respects to the one presented here, in the first half of this century. It became clear to physicists that every elementary particle sometimes has the properties of a particle and sometimes the properties of a wave. Niels Bohr proposed as a solution to this dilemma the principle of complementarity, which states that every such entity (called a particle in our language) has the properties of a wave and a particle at the same time. The nature revealed to the observer depends on the method of measurement and the quantities measured. A formal and quantitative formulation of this principle is called Heisenberg's uncertainty principle. This principle is stronger than the principle of complementarity,[7] But in this article I will only address the principle content of these determinations and not their formal quantitative form. For our purposes, this qualitative formulation of the uncertainty principle will be used: the velocity and position of a particle cannot be determined simultaneously. In other words, when a particle has a defined location, its velocity is a completely indefinite quantity that cannot be discussed and certainly not measured, and vice versa.

This principle can be linked to the arrow-in-flight paradox on several levels:

 (a) At the semantic level, according to this principle, it is not possible to formulate sentences that include the concept of velocity at the moment when there is a defined position for the particle, since these sentences contain an undefined concept: "the velocity of the particle at location x". Therefore, argument (1) leading to the conclusion that the arrow is moving at the moment it is stationary is a meaningless argument. The claim that this argument is meaningless does not rely on the law of contradiction, which states that the claim that a body is moving and not moving at the same time is invalid, but on the fact that nothing can be said about the motion of a body at the moment when its position is defined. However, formulation (2) still stands, since it is still possible to ask when the body is moving if not at the moment when it is stationary at some place. To answer formulation (2) using the principles of quantum theory, we turn to the next level:

(b) On the physical level, a particle can be found in different places at different times, even though its position is known and no velocity can be attributed to it. This stems from the fact that the concept of a moving point particle is not well defined in quantum theory. Within the framework of this theory, the particle is represented by a wave function that describes the chance of finding the particle in a particular place. The laws that describe the motion of a point object are not the laws of kinematics that relate the object's position to its velocity and acceleration through infinitesimal calculus (the Newtonian description). These laws are true for large bodies on the macroscopic level, but the dynamics of small particles are described by the laws of quantum theory (the Schrödinger equation, etc.). These laws describe the dynamics of that wave function that represents the particle. The position of the particle, which is only an average size, can be calculated from this function. At any given time, there is also a different view of the wave function that describes the probability of the particle being at a certain speed. The motion of a particle between locations is not well defined in terms of a moving point particle, but rather through the dynamics of the wave function that describes it.[8]

We would like to focus our discussion on a third, perhaps more surprising, way in which the connection between the arrow-in-flight paradox and quantum theory can be addressed:

(c) Arguments (a) and (b) above cannot provide a philosophical solution to the flying arrow problem, but only point out the connection of this problematic to that which accompanies the uncertainty principle. For this reason, we will now try to reverse the direction of reference and say that the uncertainty principle does not explain the flying arrow paradox, but is explained by it.

As is well known, the uncertainty principle has puzzled many physicists and philosophers of science in this century, and perhaps a good understanding of the arrow paradox will help to develop some insight into this principle. Of course, to do this we need to develop a good intuitive understanding of the flying arrow, in a way that is independent of quantum theory, and then try to extrapolate from it to the uncertainty principle.

The attempts to resolve the paradox by pointing out the problematic nature of the concept of continuity as presented above are not philosophically satisfactory. Therefore, I will now try to develop a further understanding of the flying arrow. I will then try to apply it back to the uncertainty principle, and then to other topics.

 

C. Does a dynamic process mean a change of static states?

 

For the purpose of the discussion, I will use a simple example from the field of elementary mechanics, although the same is true for any two concepts related to each other through the derivative in infinitesimal calculus. The concept of instantaneous velocity is defined in mechanics as the ratio between the change in place and the time period in which this change occurred, for arbitrarily small time periods. Mathematically:, where and V(t) are the position and velocity of a body at time t, respectively. This is actually the definition of the derivative of the position function with respect to time. We see that despite the fact that in order to calculate the velocity we had to observe the body over a time interval and it was not enough for us to know its position at a single moment in time, the result of the calculation is the velocity of the body at time t, which is an indivisible moment. And so, contrary to the intuitive understanding we have that says that velocity is a quantity that is defined solely over a segment of time, A body has a well-defined velocity at every indivisible instant of time.The way to calculate this velocity from the position of the body is that it forces us to use a segment of time around that instant. Of course, I do not want to claim that the body changes its position at that indivisible instant,[9] But only if it has a defined speed at that moment.

The definition of velocity through a derivative of position is called in physics the "definition" of velocity. This concept of definition indicates an operative definition (how to calculate velocity) and not a substantive definition (what velocity is).

The reason for the incorrect intuitive understanding, that speed is defined only over a period of time, is a confusion between the concept of "speed" and the concept of "change of place." The "speed" we are talking about here is the potential for a change in position, not the change itself. The change is only a consequence of the fact that a body has such a potential. From this, we can continue to argue that even if a body is in a certain place at a certain moment and at the same time also has a defined speed, this does not mean that the body is moving and not moving at the same time, as Zeno claimed. The body certainly cannot change position while having a defined position, since this is a theorem that contradicts the law of contradiction, but it does not mean that the body has no speed at that moment.[10]

To clarify the concept of velocity as potential and to distinguish it from the change in place that follows it, I will try to give two examples from the world of mechanics in which there will be no change in place even when the body has velocity. When a body collides with a wall, even if its velocity has a certain value that is different from zero, the wall will not allow it to put this potential of movement from the force into action. In other words, this is a case in which the potential (velocity) exists, but its projection (change in place) does not. In the same way, when a body collides with another body, part of the velocity it has can be transferred to the other body, so that not all of the potential will be put into action by the body that carried it before the collision. Section E will provide more examples, from other fields, of this distinction.

Having distinguished between the "velocity" which is the potential and the "change of place" which is the projection, we can add and argue that a macroscopic change of position does not necessarily entail that at every moment a microscopic change of place takes place, as is assumed at the foundation of Zeno's paradox. The change of place is a projection of the fact that the body has velocity. If we now refer to version (1) of the paradox, then we do not accept the assumption that if the arrow were not flying (in the sense of changing place) at every moment its movement would not take place. It would be more correct to say that if there were no velocity pressure at every moment its movement would not take place, and yet the fact that it has velocity at a certain moment does not contradict its being in a defined place. With reference to version (2), it is said that the question of when the arrow flies is not well defined. In both versions, the problem lies in the interpretation of the concept of "flying": if "flying" means "having velocity", the answer is that at every moment it has velocity, but if the interpretation is "changing place", it is said that the question is not well defined. The body does indeed change place throughout each interval observed, but the concept of "change of position at an indivisible moment" is not a defined concept. This is similar to the question of when time itself changes. The concept of change does not tolerate the question of when (in the sense of "at what indivisible moment"). Change always takes place over an interval. The question that can certainly be asked is whether a body has any instantaneous characteristic when its position changes. The answer to this is definitely yes, it has a velocity at every moment.

Here is the place to add the opposite aspect to this argument: the fact that "the arrow is in a certain place" does not mean that it is "not moving at that moment," since "is" does not mean "standing." It is possible to be in a certain place even in a state of motion. "Standing" means to be at zero speed, and "to be" means to be in a certain place. When we say that something is standing, it is not necessarily necessary to specify its location, which is not the case when we say that something is located, in which case we must specify where it is. This argument also leads, and perhaps more simply, to the negation of the conclusion that the arrow is Does not move While it is moving. The arrow only available At a particular place at every moment of its movement.[11]

Unlike the previous explanations, this explanation of the arrow-in-flight paradox is not related to the problematic nature of the concept of continuity. In the argument presented here, the paradox hinges on the conceptual confusion between "change of position" and "velocity." From this perspective, it is certainly possible to continue to hold to the approach that time consists of a collection of indivisible points placed densely next to each other. My intention here is not to argue for the correctness of this approach, which, as is well known, raises other problems, but only to detach it from the problematic nature of the arrow-in-flight paradox.

We will now try to delve deeper into the causes of this conceptual confusion, which, as mentioned, is rooted in the confusion between the potential of movement and its projection, which is the movement itself. When we observe[12] In a moving body, we simply observe that it is in different places at different times; we have no way of grasping the concept of speed directly without interpolation between the locations.[13] This is also the reason why the definition of velocity in mechanics is by the derivative of the position function, which is usually treated as if it were a more basic concept. Human perception is more comfortable with static concepts, and therefore it also defines dynamic concepts by using those concepts. In other words, Zeno is certainly right in that human perception cannot distinguish the velocity of a body at an indivisible moment. However, this difficulty stems from the static way in which we think. Our consciousness requires us to observe only the changes in location caused by velocity, and they of course occur only in intervals. We cannot directly observe the potential (velocity), but only its consequences. My argument here is that it cannot be concluded from this that such a velocity does not exist.

 

D. Back to the Uncertainty Principle

 

As noted in section B above (in the manner (c) in which the arrow is linked to the uncertainty principle), we can go in the opposite direction, and try to project from the understanding we have developed regarding the flying arrow to the uncertainty principle.

For this purpose, let us formulate the paradox of the arrow in flight a little differently.[14] Suppose we look at a flying arrow and photograph it at different moments (indivisible: the theoretical camera has an exposure time of 0, Ideal camera). In each photograph, the arrow will appear stationary, but its position changes each time. Now the question arises: when does it move between these different places? Our answer is that the arrow moves and at the same time has a position (and does not move and at the same time stands, as Zenon's paradoxical definition states). The fact that we do not see the movement of the arrow in the photograph must be attributed to the fact that a camera is not the appropriate device for viewing movement (or measuring speed). A camera is a device that measures (or watches) positions. We can now define another theoretical device in an analogous way: From an ideal movie. This device measures or observes the speed of a body at a single indivisible moment. As explained above, our consciousness operates in a static way, so it is difficult for us to imagine such an action. A regular film as we know it actually operates as a camera that records successive static images at a rapid rate. Our consciousness creates for us the sensation of movement by interpolating between these images. On the other hand, the ideal film is not a device that measures speed through position differences using the definition of speed as a derivative of position (as do the films in our hands, which are subject to the static limitations of our consciousness), but rather in a point-by-point manner. We will continue along the analogy and say that if we observe (record) a moving body through such a device, we can see it moving at any moment but cannot discern its position. For example, a camera with a long exposure time clearly shows that the object is moving by a trail created in the image, but a defined position cannot be discerned.[15]

We find, therefore, that the information we receive about a moving body depends on the device through which we observe it. Our consciousness, which, as mentioned, is basically static, serves us as a camera, and therefore we are directly given only information about position, while the speed is obtained indirectly by interpolation between the different positions. If we continue this line of argument, we can say that even if it were possible to know the instantaneous speed of a body by using a tape recorder, then it would not be possible to speak at the same time about its position. Now let us assume as a reasonable hypothesis that it is not possible for human consciousness to be in the mode of operation of an ideal camera and tape recorder at the same time. This theorem is in fact a simple formulation of the uncertainty principle.[16]

Here it is worth mentioning again that the uncertainty principle also includes a quantitative determination of uncertainty (Planck's constant). This, of course, cannot arise from a qualitative argument such as the one presented here, and therefore there is no pretense here of completely establishing the uncertainty principle on the basis of classical physics, but only of pointing out the fundamental element of uncertainty between pairs of dual quantities.

Another note concerning the reversal of direction from the arrow to uncertainty. The explanation of Zeno's paradox using the uncertainty principle (the semantic and physical connections above) does not require any additional assumption, whereas the explanation of the uncertainty principle using the understanding we gained from the flying arrow paradox (connection c) certainly requires an additional assumption. The discussion of the moving arrow led us to the conclusion that if a body has a definite position at a given moment, the question of when it changes its position is meaningless. But the other side of the uncertainty principle, which states that if a body has a definite velocity, the question of what its position is is meaningless, does not follow directly from the above argument. A parallel argument can be presented based on the concept of the integral (defining the position using the velocity) and shows the other side. The path I have chosen here (defining the ideal devices) seems more fruitful in the quantum context, as the reader will immediately see.

As anyone familiar with quantum theory knows, the uncertainty principle implies that there are two ways to describe dynamic quantities: either in the momentum picture (which is usually proportional to velocity), or in the position picture. These are two pictures in which the properties of the physical system are characterized by the chance of being at a certain velocity, or the chance of being at a certain location. In the first picture, all physical quantities are described using velocity (momentum) coordinates, while in the second everything is described in position coordinates. These two forms of description exclude each other (in scientific jargon: non-commutative). That is, similar to the argument presented here, the inability to know position and velocity simultaneously stems from the fact that these quantities belong to different conceptual systems (pictures) that do not "talk" to each other. We can use our terminology and say: Looking at the world through a camera gives the image of place, while looking through a film gives the image of momentum. In other words, the uncertainty between two quantities can be attributed to the fact that the two quantities belong to different conceptual worlds, similar to the description of the devices above. It was found that the argument presented here also makes sense of the dual image pairs that accompany, in quantum theory, the pairs of physical quantities that are in uncertain relationships between them.

Now an interesting objection may arise to the interpretation given here of the uncertainty principle: A commonly accepted view in the scientific community today is that the uncertainty principle does not describe us, but rather matter itself. This is not a limitation of ours, but a fundamental impossibility. Not only are velocity and position not measured simultaneously, but they are also not present (do not threaten the body) simultaneously. On the other hand, the argument presented in this article attributes uncertainty to the limitations of consciousness of the human observer. That is, according to the argument presented here, an observer who does not suffer from the limitation that consciousness cannot operate in the manner of an ideal camera and film simultaneously will be able to measure the position and velocity of a moving body simultaneously.

A possible solution to this problem, if we use Kantian terminology, is found in the distinction between the phenomenon and the noumenon. It is clear that uncertainty cannot be a property of the object as it is in itself (the noumenon), since we have no access to its properties at all. It can only be a property of the world of phenomena, that is, of the object as it appears to us (phenomenon). Although even within the world of phenomena, a distinction must be made between the object being measured as it appears to us and the measuring/observing instrument and the observer himself as they appear to us. The physical assertion that uncertainty is a property of the object and not of the measurer deals entirely with the distinction within consciousness (or within the world of phenomena). In other words, even if we agree that this is not a technological problem but a fundamental one, we can argue that this fundamental problem also originates in the properties of human perception, which defines the quantities being measured and the instruments that measure them. A more detailed discussion of this issue is a matter for a separate study.

To summarize what we have said so far: speed (or process) is a quantity that exists even in one indivisible moment, but its meaning is the potential for a change in position (or state) and not the change itself, which is only a projection of the process. The change itself, of course, cannot occur in an indivisible moment, but only over an interval. Human perception describes the process itself by the change in state caused by it, and this is because of the static nature of consciousness.[17]

E. Implications of the distinction between "process" and "change of state"

As Henri Bergson already noted in his book 'Creative Development', it is quite clear that the paradox of the flying arrow does not only concern the concept of speed, but also any process of change. In this section we will try to see the implications of the analysis presented so far for issues unrelated to mechanics.

In Section C, we defined the two concepts "process" and "change of state" in terms of potential for movement and actual movement. These two concepts, even if they are not identical, seem seemingly like twins connected together. At first glance, it seems that although we claim that these are indeed two different concepts, it is not possible to distinguish one without the other also occurring/existing at the same time. In Section C, we saw examples from the field of mechanics (collision) in which the existence of a potential can be distinguished without its actual implication.

In this section, I will attempt to discuss additional cases in which the adhesives can be separated, and discuss one separately from its counterpart. The discussion of the examples we will provide will be only cursory, and its purpose is only to provoke thought and not to provide a complete analysis of each case.

A first example would be the statement that one occasionally encounters in books on management, which states that the process of change itself is good for the organization. That is, even if there is no problem with the current structure of the organization, a benefit will arise from changing it, and this is because of the process itself, and not because the next state will be better than the current one. Of course, a change that leads to a bad static state (such as the general destruction of the machinery in a factory) does not improve the state of the organization, but if we assume that there is a group of states of the organization that are not preferable to each other, it is better for the organization to be in transition between them, rather than frozen in one of them. On the surface, it seems that we have before us a clear case of benefit arising from the existence of a process even if it is not accompanied by a change in state. It seems that to describe such benefit, we cannot use the terms of "change of states" but only those of "processes", since we do not need a different state (structure) at all, but only the process of change itself.

One might argue that this is an example that is not clear, since this statement can also be formulated in this way: there is a benefit to the organization from the very fact that its structure (state) changes, that is, that it moves from structure A to structure B. In this formulation, we have again used only the term "change of state" and have not needed the second term. It is true that it does not matter at all what the second state to which the change leads will be, but we still do not have to avoid using the term "change of state" and use only the term "process." This argument seems to be a mere semantic evasion, since to say that any other state is better, no matter which one, in fact means that it is better to be in the process of change.

Another interesting example where only one term can be used and not the other is in the discussion of the problem of God's perfection and perfection.[18] Rav Kook formulates this problem as follows: "There is a perfection of added perfection, which cannot exist in the Godhead, since the absolute, infinite perfection leaves no room for addition. And for this purpose, so that the addition of perfection also will not be lacking in being, the universal being must come into being."[19]  The basic premise is that self-improvement (i.e. spiritual progress) is one of the perfections, and therefore it must also exist in the Godhead. On the other hand, it is clear that one cannot speak of God's self-improvement for two reasons: (a) no change can occur in God; (b) God is perfect, and therefore one cannot speak of Him in terms of spiritual progress. The solution I propose here to this dilemma is parallel to that proposed for the organizations discussed above: the process itself (or at least its root) exists in God, even though the change that usually accompanies it cannot exist in Him for the reasons above. In other words, the spiritual perfection that is expressed in self-improvement is in the process and not in the change of state, and this can also characterize God.[20] It should be noted that in this example it is not possible to present an alternative formulation such as that proposed in the management example, since with regard to God it is not possible to speak at all in terms of changing states, and therefore here this is a clear case of a process that is not accompanied by a change in states at all.[21]

There are also several examples from the Talmud where this distinction between a change of state and a process can be applied. Here we will give only one example, which is discussed in more detail elsewhere,[22] And it is the giving of a get in the case of a wife's divorce. From a review of the issues that characterize the giving of a get in the Talmud, it clearly emerges that attempts to formulate the criterion for a kosher giving (that is, a giving that succeeds in applying the divorce) through "states," that is, by defining what the state is before the giving and what the state is after it, fail miserably. There are attempts to describe the giving as a transfer of ownership (the kenah of the tax) of the get, but this is hidden from several explicit laws in the Talmud. The attempt to describe the giving as a physical act of giving (the transfer of the get from the husband's hand to the wife's) also fails. The solution I proposed in my article was that the Talmud attempts to characterize a process of giving rather than a transition between states. The examples presented in the Talmud of kosher and impermissible giving of a get are very numerous. The reason is the inability to give an explicit definition of the process of giving, due to the static nature of human perception. The abundance of examples attempts to convey to the learner an intuitive feeling of what a kosher process of giving a divorce is.

F. Time in the Theory of Relativity: Bergson and Einstein

Having presented some of the implications of the distinction between the concept of "change of state" and that of "process" in fields other than physics, we will now return to another important point, again in the world of physics, where this distinction is expressed. In this section, we will discuss the description of time within the framework of Einstein's theory of relativity, and the question of whether it is exhaustive.

The most striking characteristic of time, as opposed to space or space, is that it flows. We are discussing one-dimensional space here, as there is another clear difference between space, which is three-dimensional, and time, which has one dimension. This difference is irrelevant to the discussion to be held here. Richard Taylor in his book 'Metaphysics'[23] Shows that space and time have identical characteristics, by this argument: In any sentence describing motion or a relationship between space and time, the temporal relations can be replaced by corresponding spatial relations and vice versa, and the result is always a sentence with a clear and well-defined meaning (and see various examples of this there).

From this the positivist can conclude, and indeed usually does, that time and space have identical properties. Everything that can be done in space can also be done in time, and vice versa. This is in fact also the picture obtained from Einstein's theory of relativity. In this theory, time is one of four coordinates that describe an event in Minkowski's space-time. The basic object in this theory is the "world line" of the body, that is, the line that describes the positions of the body at all times. In this picture, all of space-time is static. The flow of time is a phenomenon that physical theory completely ignores.

One of the common philosophical arguments for rejecting the sense of time passing and classifying it as a mere illusion is this argument: "A changes" means "A is in one state at one time, and in another state at another time." In particular, to say "A moves" means "A is in one place at one time, and in another place at another time."[24] Placing time as the subject (A) of one of the sentences is a meaningless statement, since time itself cannot be found in any situation/place. On time One or the other.[25]

Intuition, the sense of the passage of time, can be rescued from this predicament by claiming that the passage of time is described along a second time axis, which serves as an index that describes the state of ordinary time and its movement. But here the claim of infinite regression arises, since the new time axis will be subject to the same attack, if we want to attribute to it the same property of constant flow. This regression can be stopped by stating that there is one time that serves only as an index and does not flow, and another time that flows on the surface of the first.[26] One of the most prominent defenders of the concept of the flow of time (creative time or "duration time") was Henri Bergson in his book "Creative Development",[27] who even had a debate with Einstein who argued the opposite. Each side in this debate claimed the exclusivity of its perception, whereas I suggest here that each represented a different aspect of time, or in other words, one of the two types of time defined above. A slightly different description of the two aspects (or two types) of time was put forward by Mackaygart as early as 1908,[28] And there is even an interesting attempt to give it a certain mathematical-physical guise.[29]

Our concern here, in this description of time, is with the relevance of these two types of time to the description of processes versus changing states, as defined in the previous section. My argument is that flowing time "carries" the process, while the changing static states are characterized by an index also called time, which indicates their "position" on the timeline. This index is time of the second kind.

Here too, it seems possible to ask, in parallel with the two formulations of the arrow paradox proposed at the beginning of our discussion, how time flows. That is, at any given indivisible moment of index time (Einstein time), flowing time (Bergsonian time) is found in another time. And if so, when does it change the index? This question seems to be meaningless, since Bergsonian time is not a quantity flowing in time, but rather the flow itself. Every dynamic quantity is carried on Bergsonian time, creating a change in the static states characterized by Einstein time. Our feeling that time is passing means that we are passing on the surface of Einsteinian time, and Bergsonian time is carrying us. The implication of our flow in time is that we are each time in a time state with a different Einsteinian index.

At the beginning of my speech, I presented a definition of velocity in mechanics as a derivative of position. Later, I argued that this definition is an indirect way of deriving the process from the change of states. The use of this form stems from the limitations of human perception, which usually operates in a static receptive manner, like a camera that captures changing states and not processes. If we were able to perceive in a film-like manner, that is, the process itself, we could describe velocity directly as carried on a Bergsonian time axis. This is also the reason that Einstein's theory of relativity describes only index time, since this is the time that can be directly perceived in our consciousness. The ideal film defined in section D operates over Bergsonian time, and therefore can observe velocity (or change) at a single point in time. This is a point of Bergsonian time, which in the static perception looks like an infinitesimal.[30]

Here a question may arise, how, with regard to time, can our consciousness operate simultaneously in the manner of a camera and a film, and perceive both types of time. Two lines of thought can be presented to answer this question. First, it seems more intuitive to say that, with regard to time itself, our direct experience is precisely the dynamic one, not the static one. We feel time flowing, and artificially mark static index points on it. And perhaps this is itself the essence of the difference between space and time, that one is perceived only statically, and the other only dynamically. Another line of thought may arise here, and that is that the perception of Einsteinian and Bergsonian time is made at two different levels of consciousness. Einsteinian time seems to have more objective (or intersubjective) significance, while Bergsonian time is perceived as more subjective. Naturally, science describes our "objective" experiences, and therefore in physics the perception of time as an index dominates. In any case, the assertion that our consciousness, acting as a camera, cannot perceive processes directly but only through changes in state, refers to objective consciousness that can be quantified mathematically. I do not wish to argue against the existence of an immediate experience of our consciousness experiencing the passage of time.[31]

[1] I would like to thank Amnon Levav, Dalia Derai, and Ladi Prudovsky for their careful reading of this article, and for their helpful comments that contributed greatly to its writing. I would also like to thank Avshalom Elitzur for a discussion I had with him in the very early stages of formulating the ideas presented here, and for reading the almost final version of the article.

[2]   cf. F. Cajori, "History of Zeno's Arguments Against Motion", American Mathematical Monthly, 22 (1915); WC Salmon, 'Zeno's Paradoxes', Indianapolis, Bobbs-Merrill, 1970.

[3] This formulation is presented in Shmuel Hugo Bergman, 'Introduction to the Theory of Logic', Jerusalem, Bialik Institute, 1975, in the third chapter, section 18.

[4] This approach is more characteristic of the Leibnizian formulation of the differential calculus, as opposed to the less intuitive Newtonian formulation. For an intuitive formulation of the differential calculus, see HJ Keisler, 'Foundations of infinitesimal calculus', Boston, Prindle, c1976.

[5] There are attempts to present formal solutions to the arrow paradox by assuming a discrete structure of space and time. For a more detailed discussion, see WC Salmon, Space, Time and Motion, Dickenson, 1975

See also the book mentioned in note 2 by the same author.

[6] My thanks to an anonymous reviewer of this article who drew my attention to the approach of synthetic differential geometry in which the continuum is not composed of points. See, for example, J. L. Bell, “Infinitesimals and the Continuum”, Mathematical Intelligencer 17 (1995): 55-57; idem, “Infinitesimals”, Synthesis 75 (1988): 285-315

Although the logic underlying this approach is intuitionistic, as argued in the body of the article, common sense still views the concept of a point in time as evidential.

[7] This principle also determines a ratio between the uncertainty measures of the two complementary properties, beyond the principled determination that it is not possible to know these two values simultaneously.

[8] Zenon's paradox can of course be extended to any situation in which there is a change (see below at the beginning of section e), and not necessarily to the concept of velocity. In this way, the dynamic laws describing the variability of the wave function can be made difficult by the same question that Zenon made difficult for the flying arrow: the function changes "position" (in the space of functions) over time. If so, at every indivisible moment of time the function is changing and at the same time static. A discussion of this question involves a discussion of the ontological status of the wave function, an issue which, as far as I know, is far from being understood.

[9] When it moves at a finite speed, this is obvious. But even when its speed is infinite, it seems that it is impossible to change position in one indivisible moment of time. In this case, we can change position in a short segment of time as we wish, but not in an indivisible moment. A point body cannot be in two places at the same indivisible moment of time: this is a logical contradiction and not a physical inability to reach infinite speed.

[10] In field theory in physics, one sometimes deals with an ultrastatic and ultralocal model. This is a theory that describes a field defined over a single indivisible moment of time, and a single indivisible spatial point. A velocity (or momentum) field can be defined in such a model. It is clear that if velocity is a change in location over time, this field has no meaning within the framework of such a model.

[11] See a similar discussion in Salmon's books mentioned in note 5. In contrast to the discussion presented here, he does seem to link this argument to problems with the concept of continuity.

[12] "Observers" is used in this paragraph to describe cognition, not perception. I am aware that the distinction between them is not always sharp.

[13] This claim is one of the pillars of the argument in Henri Bergson's book 'Creative Development', Jerusalem, Magnes, 1978. He even uses it there in the context of the discussion of Zeno's Flying Arrow. See also section 6 of Lakman.

[14] See also Bergson's discussion 'Creative Development', Chapter 4.

[15] An example of this kind was given in Aharon Pinker's book, 'The Book of the Atom', Jerusalem, Reuven Mass. My thanks to Itamar Pitovsky of the Hebrew University for bringing this source to my attention.

[16] For those who do not feel the plausibility of this hypothesis, we can change the direction of the argument again here, and use the uncertainty principle as support for the hypothesis that states that consciousness cannot simultaneously act in the manner of an ideal camera and film.

[17] It is possible to say that speed and change of place (or, in general, the process and the change it causes) are not two things that exist on the same plane. The process is the thing in itself, while the change that follows it is the phenomenon. If this statement is correct, then it is not possible to speak of these two quantities as dual in such a way that one is captured by a camera and the other by a film. The thing in itself cannot be captured by any device. A detailed analysis of this statement raises several problems that are beyond the scope of this article, and therefore such a relationship has not been established in the body of the article.

[18] For a discussion of this problem, see Yosef ben Shlomo, "Completion and Completion in the Teachings of the Godhead of Rabbi Kook," 'Iyun' 33, (5784): 289.

[19] Rabbi Kook, Orot Hakodesh, Jerusalem, Rabbi Kook Institute, 1985, volume 2, page 171.

[20] The solution offered there by Rabbi Kook himself seems, on the surface, different. In my opinion, his solution is very close to the one offered here, but that is a matter for a separate discussion.

[21] We will note here that if we adopt the view we presented in note 17, according to which the "process" is the thing as it is in itself of the "change", then in God the thing is found as it is in itself in such a way that its appearance to the eyes of consciousness (the change) is expressed in creatures. This is a panentheistic view that is more similar to that proposed by Rabbi Kook there. See Ben Shlomo's article cited in note 18 above.

[22] See Michael Avraham, Weekly Page, Department of Basic Studies at Bar-Ilan University, Parashat Ki Titze, 1995, Torah and Science Section.

[23] Translated into Hebrew by Adam Publishing, Free University, 1983. See there, Chapter 7.

[24] See Avshalom Elitzur, 'Time and Consciousness', Tel Aviv, Broadcasting University Press, 1994, Chapter 4.

[25] In the second formulation, space cannot be placed as the subject of the sentence. This stems only from the fact that we have chosen to discuss an example of change, that is, movement, and not the concept of change itself as in the first formulation. The general sentence is the first formulation, the second is only a concrete example intended to illustrate.

[26] One can speak of two aspects of time rather than two different times. Mactagart's formulations (see note 28) seem closer to such a formulation.

[27] See note 13.

 [28] Philosophical Studies, ed. SV Keeing, London, E. Arnold, 1934, chap. 5; , McTaggart JE

And the place number 30 in the article by Horvitz Arshansky and Elitzur appears in the following note.

[29] LP Horwitz, RI Arshansky and AC Elitzur, "On the Two Aspects of Time: The Distinction and Its Implications", Foundations of Physics, 18, 12 (1988): pp. 1159-1193

[30] Berkeley's statement regarding the concept of the infinitesimal is well known: "ghosts of recently departed quantities" ("The Analyst", reprinted in James R. Newman' ed. The World of Mathematics, New York, Simon and Schuster, 1956); cf. notes 6

[31] This aspect is reminiscent of Schopenhauer's view that the person looking inward into himself can perceive the thing as it is for himself (the soul/spirit), whereas in the direct perception of objects outside himself he is of course limited to the phenomenon only (the body). Physics describes the intersubjective aspects of perception, and is therefore limited to the description of index time only. Philosophy (in certain of its conceptions) and mysticism are of course freed from these shackles, and therefore these disciplines often rebel against the difficulties of scientific perception.